A. Borichev, R. Dhuez, K. Kellay
We obtain sampling and interpolation theorems in radial weighted spaces of analytic functions for weights of arbitrary (more rapid than polynomial) growth. We give an application to invariant subspaces of arbitrary index in large weighted Bergman spaces.
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We extend our work on nonseparated interpolating sequences, originally developed for Bergman spaces with weights of the form $(1 - |z|^2)^α$, to more general weights.
An approach to complex interpolation of compact subsets of $\Bbb C^n$, including Brunn-Minkowski type inequalities for the capacities of the interpolating sets, was developed recently by means of plurisubharmonic geodesics between relative extremal functions of the given sets. Here we show that a much better control can be achieved by means of the geodesics between weighted relative extremal functions. In particular, we establish convexity properties of the capacities that are stronger than those given by the Brunn-Minkowski inequalities.
We use $L^2$ estimates for the $\bar\partial$ equation to find geometric conditions on discrete interpolating varieties for weighted spaces $A_p(\C)$ of entire functions such that $| f(z)|\le Ae^{Bp(z)}$ for some $A,B>0$. In particular, we give a characterization when $p(z)=e^{| z|}$ and more generally when $\ln p(e^r)$ is convex and $\ln p(r)$ is concave.
It is well known from a result by Shapiro-Shields that in the Hardy spaces, a sequence of reproducing kernels is uniformly minimal if and only if it is an unconditional basis in its span. This property which can be reformulated in terms of interpolation and so-called weak interpolation is not true in Paley-Wiener spaces in general. Here we show that the Carleson condition on a sequence $Λ$ together with minimality in Paley-Wiener spaces $PW_τ^{p}$ of the associated sequence of reproducing kernels implies the interpolation property of $Λ$ in $PW_{τ+ε}^{p}$, for every $ε>0$. With the same techni…
We study those smooth complex hypersurfaces W in C^n having the property that all holomorphic functions of finite weighted L^p norm on W extend to entire functions with finite weighted L^p norm. Such hypersurfaces are called interpolation hypersurfaces. We also examine the dual problem of finding all sampling hypersurfaces, i.e., smooth hypersurfaces W in C^n such that any entire function with finite weighted L^p norm is stably determined by its restriction to W. We provide sufficient geometric conditions on the hypersurface to be an interpolation and sampling hypersurface. The geometric condi…
We present sufficient conditions on a smooth uniformly flat hypersurface W in the unit ball to be an interpolation hypersurface or a sampling hypersurface for generalized Bergman spaces associated to the unit ball with its Bergman metric. The conditions are phrased in terms of a geometric Beurling-type densities, similar to the densities defined for hypersurfaces in C^n by Ortega-Cerda, Schuster and the second author. For the case of sampling, our proofs are different than those in the C^n case; we use an approach closer in spirit to the work of Berndtsson and Ortega-Cerda in the 1-dimensional…
We study multiple sampling and interpolation problems with unbounded multiplicities in the weighted Bergman space, both in the hilbertian case p = 2 and the uniform case p = +$\infty$.
We prove that square integrable holomorphic functions (with respect to a plurisubharmonic weight) can be extended in a square integrable manner from certain singular hypersurfaces (which include uniformly flat, normal crossing divisors) to entire functions in affine space. This provides evidence for a conjecture regarding the positivity of the curvature of the weight under consideration.
We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight $λ$ depends on the independent variable $z$. We prove that for an increasing radial weight $λ(z)$ the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight $λ(z)$ we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial s…
We study the uniqueness sets, the weak interpolation sets, and convergence of the Lagrange interpolation series in radial weighted Fock spaces
The Patterson-Sullivan construction is proved almost surely to recover a Bergman function from its values on a random discrete subset sampled with the determinantal point process induced by the Bergman kernel on the unit ball $\mathbb{D}_d$ in $\mathbb{C}^d$. For super-critical weighted Bergman spaces, the interpolation is uniform when the functions range over the unit ball of the weighted Bergman space. As main results, we obtain a necessary and sufficient condition for interpolation of a fixed pluriharmonic function in the complex hyperbolic space of arbitrary dimension (cf. Theorem 1.4 and …
Two weights inequality for Hankel operators on weighted Bergman spaces induced by radial weights
本文在正则假设下研究了径向权重诱导的加权Bergman空间上Hankel算子的双权不等式问题，刻画了该算子的有界性与紧致性。
We consider a Nevanlinna-Pick interpolation problem on finite sequences of the unit disc D constrained by Hardy and radial-weighted Bergman norms. We find sharp asymptotics on the corresponding interpolation constants. As another application of our techniques we prove embedding theorems for rational functions. We find that the embedding of H $\infty$ into Hardy or radial-weighted Bergman spaces in D is invertible on the subset of rational functions of a given degree n whose poles are separated from the unit circle and obtain asymptotically sharp estimates of the corresponding embedding constan…
Answering a question of Lindholm, we prove strict density inequalities for sampling and interpolation in Fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight. In particular, these spaces do not admit a set that is simultaneously sampling and interpolating. To prove optimality of the density conditions, we construct sampling sets with a density arbitrarily close to the critical density. The techniques combine methods from several complex variables (estimates for $\bar \partial$) and the theory of localized frames in general reproducing kernel Hilbert…
The purpose of this paper is to prove an interpolation formula involving derivatives for entire functions of exponential type. We extend the interpolation formula derived by J. Vaaler in [37, Theorem 9] to general $L^p$ de Branges spaces. We extensively use techniques from de Branges' theory of Hilbert spaces of entire functions as developed in [6], but a crucial passage involves the Hilbert-type inequalities as derived in [15]. We give applications to homogeneous spaces of entire functions that involve Bessel functions and we prove a uniqueness result for extremal one-sided band-limited appro…
In this paper we completely characterize those weighted Hardy spaces that are Poletsky--Stessin Hardy spaces $H^p_u$. We also provide a reduction of $H^\infty$ problems to $H^p_u$ problems and demonstrate how such a reduction can be used to make shortcuts in the proofs of the interpolation theorem and corona problem.
A pluripotential theoretic framework for polynomial interpolation of vector-valued functions and differential forms
本文提出修正的加权多势理论框架，引入向量值权重，将标量值函数插值的渐近结果部分推广到有限维复内积空间中的向量值函数，并应用于微分形式的多项式系数插值。
Summing Carleson measure on radial weighted Begman spaces
本文研究了径向加权Bergman空间上正Borel测度诱导的Carleson嵌入的r-求和性质，给出了对所有0<p,q<∞成立的条件。
Carleson measures and interpolating and sampling sequences for weighted Bergman spaces on the unit disk are described for weights that are radial and grow faster than the standard weights $(1-|z|)^{-α}$, $0<α<1$. These results make the Hardy space $H^2$ appear naturally as a "degenerate" endpoint case for the class of Bergman spaces under study.
Strong and weak-type estimates for radial weighted Bergman projections
研究单位圆盘上由径向权重诱导的伯格曼投影的\(L^p\)有界性，通过建立二分法及相关条件刻画，证实佩莱兹 - 拉蒂亚猜想，解决多斯塔尼奇在2004年提出的问题。
Hypercyclicity of weighted shifts on weighted Bergman spaces
本文针对加权Bergman空间上的加权后移位算子，基于系数泛函范数估计分析其连续性与动力学性质，给出径向权情形下超循环移位的完整刻画，得到了新的系数乘子结果。
Suppose that $ω$ is a radial weight on the unit disk that satisfies both forward and reverse doubling conditions. Using Carleson measures and $T1$-type conditions, we obtain necessary and sufficient conditions of the positive Borel measure $μ$ such that the Toeplitz operator $T_{μ,ω}:L^p_a(ω)\to L_a^1(ω)$ is bounded and compact for $0<p\leq 1$. In addition, we obtain a bump condition for the bounded Toeplitz operators with $L^1(ω)$ symbol on $L^1_a(ω)$. This generalizes a result of Zhu in \cite{zhu1989}.
This paper concerns a commutant lifting theorem and a Nevanlinna-Pick type interpolation result in the setting of multipliers from vector-valued Drury-Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $\mathbb{C}^n$. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.
For $α>-1$ and $0<p<\infty$, we study weighted Bergman spaces $\mathcal B^p_α$ of harmonic functions on the real hyperbolic ball and obtain an atomic decomposition of these spaces in terms of reproducing kernels. We show that an $r$-separated sequence $\{a_m\}$ with sufficiently large $r$ is an interpolating sequence for $\mathcal B^p_α$. Using these we determine precisely when a Bergman space $\mathcal B^p_α$ is included in another Bergman space $\mathcal B^q_β$.
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